Nouveaux paradigmes en dynamique de populations hétérogènes : modélisation trajectorielle, agrégation, et données empiriques

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hétérogènes : modélisation trajectorielle, agrégation, et

données empiriques

Sarah Kaakai

To cite this version:

Sarah Kaakai. Nouveaux paradigmes en dynamique de populations hétérogènes : modélisation trajectorielle, agrégation, et données empiriques. Probability [math.PR]. Université Pierre et Marie Curie -Paris VI, 2017. English. �NNT : 2017PA066553�. �tel-01897474�

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56

THÈSE DE DOCTORAT DE L’UNIVERSITÉ PIERRE ET

MARIE CURIE

Spécialité : Mathématiques Appliquées

École doctorale de Sciences Mathématiques de Paris-Centre

Présentée par

Sarah KAAKAI

Nouveaux paradigmes en dynamique de populations

hétérogènes :

Modélisation trajectorielle, Agrégation, et Données empiriques

New paradigms in heterogeneous population dynamics:

Pathwise modeling, aggregation, and empirical evidence

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Résumé

Cette thèse porte sur la modélisation probabiliste de l’hétérogénéité des populations humaines et de son impact sur la longévité. Depuis quelques années, de nombreuses études montrent une augmentation alarmante des inégalités de mortalité géographiques et socioéconomiques. Ce changement de paradigme pose des problèmes que les modèles démographiques traditionnels ne peuvent résoudre, et dont la formalisation exige une ob-servation fine des données dans un contexte pluridisciplinaire. Avec comme fil conducteur les modèles de dynamique de population, cette thèse propose d’illustrer cette complexité selon différents points de vue: Le premier propose de montrer le lien entre hétérogénéité et non-linéarité en présence de changements de composition de la population. Le processus appelé Birth Death Swap est défini par une équation dirigée par une mesure de Poisson à l’aide d’un résultat de comparaison trajectoriel. Quand les swaps sont plus rapides que les évènements démographiques, un résultat de moyennisation est établi par convergence stable et comparaison. En particulier, la population agrégée tend vers une dynamique non-linéaire. Nous étudions ensuite empiriquement l’impact de l’hétérogénéité sur la mortalité agrégée, en s’appuyant sur des données de population anglaise structurée par âge et circonstances socioéconomiques. Nous montrons par des simulations numériques comment l’hétérogénéité peut compenser la réduction d’une cause de mortalité. Le dernier point de vue est une revue interdisciplinaire sur les déterminants de la longévité, accompagnée d’une réflexion sur l’évolution des outils pour l’analyser et des nouveaux enjeux de modélisation face à ce changement de paradigme.

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Abstract

This thesis deals with the probabilistic modeling of heterogeneity in human populations and of its impact on longevity. Over the past few years, numerous studies have shown a significant increase in geographical and socioeconomic inequalities in mortality. New issues have emerged from this paradigm shift that traditional demographic models are not able solve, and whose formalization requires a careful analysis of the data, in a multidisciplinary environment. Using the framework of population dynamics, this thesis aims at illustrating this complexity according to different points of view: We explore the link between heterogeneity and non-linearity in the presence of composition changes in the population, from a mathematical modeling viewpoint. The population dynamics, called Birth Death Swap, is built as the solution of a stochastic equation driven by a Poisson measure, using a more general pathwise comparison result. When swaps occur at a faster rate than demographic events, an averaging result is obtained by stable convergence and comparison. In particular, the aggregated population converges towards a nonlinear dynamic. In the second part, the impact of heterogeneity on aggregate mortality is studied from an empirical viewpoint, using English population data structured by age and socioeconomic circumstances. Based on numerical simulations, we show how a cause of death reduction could be compensated in presence of heterogeneity. The last point of view is an interdisciplinary survey on the determinants of longevity, accompanied by an analysis on the evolution of tools to analyze it and on new modeling issues in the face of this paradigm shift.

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A mes parents Hafid et Sylvie, A Céline,

No man is an island, Entire of itself; Every man is a piece of the continent, A part of the main. John Donne

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List of figures

1.1 Age pyramid of metropolitan France (Source: INSEE) . . . 7

1.2 Example of distribution of swap events and demographic events . . . 23

1.3 Age pyramids in 2001 and 2015 (Figure 4.1) . . . 34

1.4 Proportion of males by age class and IMD quintile (Figure 4.2) . . . 36

1.5 Males averaged annual improvement rates (Figure 4.12) . . . 40

3.1 Example of distribution of swapevents and demographicevents . . . 74

4.1 Age pyramids in 2001 and 2015 . . . 107

4.2 Proportion of males by age class and IMD quintile . . . 109

4.3 Proportion of males by cohort and IMD quintile . . . 110

4.4 Central death rates per single year of age and IMD quintile in 2015 . . . 111

4.5 Average annual rates of improvement in mortality, 1981-2015 . . . 112

4.6 Average annual rates of improvement in mortality, males . . . 113

4.7 Males deaths per cause and IMD quintile for ages 65-85 . . . 114

4.8 Life expectancy at age 65 over 1981-2015 . . . 115

4.9 Aggregated age pyramid with initial population based on the 1981 data . 126 4.10 Aggregated age pyramid with initial population based on the 2015 data . 126 4.11 Evolution of males life expectancy at age 65 . . . 128

4.12 Males average annual mortality improvement rates . . . 129

4.13 Life expectancy over time . . . 132

4.14 Aggregated period life expectancy over time . . . 134

4.15 Aggregated period life expectancy over time: Ischemic heart diseases reductions . . . 135

4.16 Aggregated period life expectancy over time . . . 136

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B.2 IMD 2015 quintile proportions for the Cohort 1921-1930 . . . 142

B.3 Central death rates per IMD 1y 1981, 2015 . . . 142

B.4 Females improvement rates . . . 143

B.5 Deaths per cause and IMD quintile for females of age in 45-65 . . . 144

B.6 Deaths per cause and IMD quintile for males of age in 45-65 . . . 145

B.7 Life expectancy at age 25 over 1981-2015 . . . 146

D.1 Evolution of female life expectancy at age 65 . . . 151

D.2 Females averaged annual improvement rates . . . 152

D.3 Aggregated period life expectancy over time . . . 152

D.4 Aggregated life expectancy over time . . . 153

D.5 Aggregated period life expectancy over time with birth and mortality changes . . . 153

5.1 Preston curves, 1900, 1930, 1960, reproduced from Preston (1975) . . . . 165

5.2 Preston curve in 2000, reproduced from Deaton (2003) . . . 165

5.3 Relationship between median county income and standardized mortality rates among working-age individuals, reproduced from Wilkinson and Pickett (2009) (Figure 11) . . . 175

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Chapter 1 Introduction

In over two centuries, the world population has been transformed dramatically, under the effect of considerable changes induced by demographic, economic, technological, medical, epidemiological, political and social revolutions. The age pyramids of ageing developed countries look like “colossus with feet of clay”, and the complexity of involved phenomena make the projection of future developments very difficult, especially since these transitions are unprecedented.

The problem does not lie so much in the lack of data or empirical studies. For several years now, a considerable amount of data have been collected at different levels. A number of international organizations1 _{have their own open databases, and national}

statistical institutes2 _{have been releasing more and more data. On top of that, more}

than fifty public reports are produced each year. The private sector is also very active on these issues, especially pension funds and insurance companies which are strongly exposed to the increase in life expectancy at older ages.

However, the past few years have been marked by a renewed demand for more efficient models. This demand has been motivated by observations of recent demographic trends which seem to be in contradiction with some firmly established ideas. New available data seem to indicate a paradigm shift over the past decades, toward a more complex and individualized world. Countries which had similar mortality experiences until the 1980s now diverge, and a widening of health and mortality gaps inside countries has been reported by a large number of studies. These new trends have been declared as key public issues by several organizations, including the WHO in its latest World report on ageing

1_{For instance, the United Nations (UN) (unstats.un.org), the World Bank (WB) (data.worldbank.org)}

or the World Health Organisation (WHO) (who.int/gho).

2_{such as the Institut National de la Statistique et des Etudes Economiques (INSEE) and the Institut}

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and health (World Health Organization (2015)), and the National Institute on ageing in the United States, which created in 2008 a panel on Understanding Divergent Trends in Longevity in High-Income Countries, leading to the publication of a comprehensive report in National Research Council and Committee on Population (2011). There is thus an important need for finer grained models capable of integrating the population heterogeneity, interactions at several scales or variability of the environment.

In the first part of the introduction of this thesis, we first give a general background on the evolution of human longevity. In the second part, we give a more detailed description of the issues which motivated this thesis, followed by an outline of the different approaches used in this thesis. A summary of the results of the thesis is given in the last part of the introduction.

1.1 General background

As stated by Vaupel and coauthors in their famous 2009 article, “The remarkable gain of

about 30 years in life expectancy in western Europe, the USA, Canada, Australia, and New Zealand—and even larger gains in Japan and some western European countries, such as Spain and Italy—stands out as one of the most important accomplishments of the 20th century ” (Christensen et al.(2009)). For more than 150 years, record female life expectancy has grown at the pace of almost 2.5 years per decade (Oeppen and

Vaupel (2002)). In France, the investigations of Louis Henry in the field of Historical

Demography have allowed the construction of historical mortality tables going back to the mid-eigteenth century, thanks to the remarkable work ofVallin and Meslé (2001). Based on their estimations, life expectancy at birth was estimated to be about be 25 years of age in the mid-eighteenth century. By 1830, it had attained 40 years and remained stable until the 1870s. Less than a century later, in 1950, life expectancy had increased of more than 30 years, to 69 years for females and 63 for males. Today, life expectancy in France is estimated to have reached the age of 85 for females, and 79 for males3_{. Obviously,}

these dramatic changes have been accompanied by major societal and economic changes, and the future consequences of such levels of longevity are the source of numerous and urgent debates, both in the research community and in the civil society.

The demographic transition The sustained improvement of the duration of life, together with fertility decline, is part of the larger process of the demographic transition,

3_Source: _{Cambois et al.} ₍₂₀₀₉_{), Institut National de la Statistique et des Etudes Economiques}

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1.1 General background

which has been ongoing in most developed country since the mid-eighteenth century. These major demographic changes have been both the source and the consequence of massive social and economic upheavals which include the industrial revolution, rampant urbanization, the increase of living standards and educational levels, together with greater social and political equality, especially between men and women with the entry of women in the labor force. The demography transition was “the most important social and

economic change to take place in Europe in centuries” (Reher (2011)). In countries which experienced the historic transition4_{, important public health and medical advances have}

significantly contributed to mortality declines during the twentieth century5 ₍_{Cutler et al.}

(2006)). At the same time, the reduction of mortality at younger ages, which took place before the beginning of fertility declines, led to a spectacular growth of the population and the increase of the proportion of individuals of working-age until some time between the late 1950s and the early 1980s. Consequently, these economically advantageous age-structures have contributed to create favorable conditions for the development of national pension schemes, in which the large working-age population could pay pension benefits for a small group of elderly people (Reher(2011)). In turn, the improvement of social welfare has probably contributed to further mortality declines.

New challenges brought by the demographic transition But today, the impact of longevity improvements is also producing new issues and challenges at multiple societal levels. Population ageing poses a major challenge to the sustainability of intergenerational risk sharing mechanisms such as pay-as-you-go systems and public health systems. In the recent years, pension funds and governments have been closing Defined Benefit retirement plans or shifting them toward a Defined Contribution system6_{. This shows an indicator}

of a transfer of the demographic risk back to the policyholders (Barrieu et al. (2012)). As also stated in Barrieu et al. (2012), the insurance industry is facing challenges linked to increasing longevity, in the form of greater regulatory capital and the need to transfer part of their risk to reinsurers and financial markets. The increase in longevity is also generating a whole new paradigm in the way the life course is perceived, and the timescale of many life course strategies have deeply changed. The traditional three broad periods of life (childhood, adulthood and old age) have been replaced by a four period model, with the third age period being divided in two stages, “young old” and “oldest old”, the

4_{The historic transition affected most of European countries and countries with European roots}

(Argentina, Uruguay, the United States, Canada, New Zealand (Reher(2011)).

5_{We will return to the demographic transition and the importance of public health in Chapter 5.} 6 _{A Defined Benefit retirement plan means that pension benefits are predetermined, based on}

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latter being characterized by reduced autonomy and greater medical needs (Christensen

et al.(2009)). The notion of age itself has changed. At a fixed age, individuals seem to

have rejuvenated: individuals now “become old at older ages” than before (d’Albis and

Collard (2013)). As retirement age has been raised in several countries, there is also a

need to redefine and redistribute work within these populations which are simultaneously ageing and rejuvenating, as argued in Vaupel and Loichinger (2006).

An heterogeneous evolution of longevity If the increase in record life expectancy has been rather stable, a more detailed analysis of underlying processes is by no means simple. Historically, the decline in mortality was not uniform in age, and while life expectancy at birth increased by 30 years in England between 1841 and 1950, life expectancy at age 10 increased of only 15 years and life expectancy at 65 remained virtually constant until 1950. Studies on socioeconomic inequalities in mortality, which is one of the main topics of this thesis, can be traced back to the early eighteenth century. For instance, Villermé (1830) was one of the first people to exhibit a link between mortality rates and socioeconomic status7_{. More recently a growing amount of evidence}

seems to indicate that along with the increase of life expectancy, developed countries have also experienced a widening of socioeconomic gaps in health and mortality since the second half of the twentieth century (Cutler et al.(2006); Elo (2009); National Research

Council and Committee on Population(2011); Olshansky et al. (2012)). For instance,

Olshansky et al.(2012) in the United States have found out that the life expectancy gap

(at birth) between females with less than 12 years of education and females with more than 16 years of education grew from 7.7 years in 1990 to 10.3 years in 2008 (13.4 to 14.2 for males). Life expectancy of less educated white females and males even decreased during this 18-year period.

The pervasive effect of socioeconomic factors on health and mortality is also at the source of economic and social issues, and has become an important public problem for many countries and international organizations. For instance the World Health Organization (WHO) named the reducing of inequities one of the key issues for public health action in their last report on ageing and health (World Health Organization (2015)). Beside reducing the fundamental inequity in the correlation of the duration of life with one’s income or social status, they recommended to target policies overcoming these inequities, in the sense that “strategies must look not just improve conditions for the best-off or

the average older person. Attention must also be given to [..] narrowing the gaps in

7_{For this purpose, he studied deaths in Paris per borough and did what is now called a data linkage}

with tax authorities statistics, by comparing mortality rates to rates of non-taxable households per borough (seeMireaux(1962) for more details).

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1.1 General background

the total inequalities observed among older individuals”. To that matter, understanding

the social determinants at the root of these inequalities is critical for designing efficient policies. However, underlying factors responsible for socioeconomic differences in health and mortality are still not clearly understood. As a consequence, policy recommendations can differ substantially, according to the theory taken into account in order to model pathways involved into translating socioeconomic status into mortality outcomes. These issues are discussed in more detail in Chapter 5.

Socioeconomic differences in mortality can also impact the equity of national pension schemes by allocating more resources to individuals higher in the socioeconomic ladder, already benefiting from better health and longevity (Villegas and Haberman (2014)). Furthermore, not taking heterogeneity into account might lead to significant errors when assessing pensions liabilities or regulatory capital. A striking and simple example is the evolution of the socioeconomic composition of cohorts of seniors. As individuals in the same cohort grow older, low-income pensioners die at earlier ages, so that average benefits increase and costs rise (Edwards and Tuljapurkar (2005)). This phenomenon is typically nonlinear, and is one of many issues generated by the interactions between the evolution of an heterogeneous population and aggregate indicators, to which we will come back to in Chapter 4.

Two centuries of interdisciplinary literature All these challenges have generated a considerable amount of research, producing an interdisciplinary literature in fields ranging from mathematics to history, and also including actuarial science, biodemography, biology, computer science, demography, economics, epidemiology, medical research, public health or sociology8_{. In 1825, B. Gompertz presented to the Royal Society of London}

his “ law of human mortality”, describing age specific mortality rates as an exponential function of age, with only two parameters (Gompertz(1825), see alsoKirkwood(2015) for a commentary on Gompertz’s original article). Gompertz, who was an English actuary, was actually interested in improving the calculation of rates for the selling and purchasing of annuities. Almost two centuries later, his law of mortality is still regarded as a kind of “fundamental law of mortality”, and seems to hold for a wide range of species. On the other hand, a justification for the existence of this law has not been widely established yet. This illustrates how little we still know on the evolution of human longevity and its societal impacts, as many questions are still open and the source of heated debates. In the new context of open data, population data have also been increasingly released by

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governmental statistical institutions and international agencies. For instance, The United Nations, the World Bank or the WHO now all have their own online open database. These data allow new questions to be explored and shed a new light on issues at stake. Thus, the multiplicity of approaches and data accounts for both the richness and the difficulty of this subject. This thesis lies within the broader scope of understanding these approaches and their link to social issues, in order to provide a theoretical and simulation framework to compare and question some of the common practices. In the remainder of this section, we will detail more precisely the questions which served as guidelines for this thesis, and then move to a brief description of the different approaches used to address the concerned issues.

1.2 Motivations of the thesis

What can population dynamics do for longevity ? The aggregate longevity observed at the macroscopic level of a national population is the result of complex non-linear demographic mechanisms. In presence of heterogeneity, significant longevity variations could be induced by changes in the cohorts’ composition or size, caused for instance by changes in the fertility processes or interactions between individuals. Even the estimation of quantities such as annual death probabilities can be a complex task. Usually, these quantities are estimated on samples as large as possible, due to the rare occurrence of death events. However, an increase in the sample population size might also mean an increase of its heterogeneity. The more heterogeneous a population is, the further from average may individuals behave, thus increasing the variance of estimators and creating a trade-off on the population size.

Standard statistical mortality models such as the Lee-Carter model and its extensions

(Lee and Carter (1992); Renshaw and Haberman (2006)) or the Cairns-Blake-Dowd

(CBD) model (Cairns et al. (2006)) are based on the modeling of age-specific mortality rates as time series, in order to make projections of future mortality rates. More recently, a new modeling method for mortality rates has been proposed byLudkovski et al.(2016), based on Gaussian Processes models, and able to quantifies uncertainty associated with smoothed historical experience. The dramatic changes in the demographic and societal structure of populations in developed countries question however the ability of historic data to be a “good guide to the future”. In Metropolitan France for instance, the population rose from 41.7 millions in 1950 to 64 millions in 2014, an increase of over 54%. 11% of the population was more 65 years old in 1950, in comparison with over 18%

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1.2 Motivations of the thesis

today9_{.The age pyramids of Metropolitan France in 1950 and 2014 are represented in}

Figure 1.1. Those evolutions have changed the way seniors are perceived, which in turn might have influenced the evolution of their mortality. The composition of the population has also changed significantly. For instance, the proportion of individuals of age 30-45 with no diploma or only a primary school diploma has dropped from about 40% in 1980 to less than 15% in 201010_{. There is thus an inherent complexity in comparing inside the}

same time series populations so different, in order to make robust projections.

Furthermore, insurers or pensions funds are mainly interested in mortality at older ages, typically above 65, and sometimes consider these ages only in their models. Limiting data to this age class constitutes however a substantial loss of information, by failing to capture information on younger cohorts (social composition, smoking habits...) and which can give valuable insights into the future, since “ today’s youths are tomorrow’s seniors”. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100+ 1950 1945 1940 1935 1930 1925 1920 1915 1910 1905 1900 1895 1890 1885 1880 1875 1870 1865 1860 1855 1850 400 300 200 100 0 100 200 300 400

Number of individuals (in thousands)

Age

Y

ear of bir

th

Type Males Females

(a) 1950 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100+ 2014 2009 2004 1999 1994 1989 1984 1979 1974 1969 1964 1959 1954 1949 1944 1939 1934 1929 1924 1919 1914 400 300 200 100 0 100 200 300 400

Number of individuals (in thousands)

Age

Y

ear of bir

th

Type Males Females

(b) 2014

Fig. 1.1 Age pyramid of metropolitan France(Source: INSEE)

Heterogeneous population dynamics For these reasons, this thesis focuses on the probabilistic modeling of heterogeneous population dynamics rather than on the statistical projection of mortality rates, by taking on an approach similar to that of N.

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El Karoui and coauthors (Bensusan (2010) and Boumezoued(2016)). We are interested in the mathematical modeling of complex population dynamics, as an experimenting and simulation tool to generate scenarios, rather than to make realistic predictions.

As for all human systems, the study of human population dynamics is complex due to the very nature of underlying mechanisms. Phenomena are often non-stationary, heterogeneous, and often include interactions taking place at different scales and with sometimes opposite effects. Due to these difficulties, producing a pertinent modeling directly at the macro-level appears to be a very complicated, if not impossible, task. Hence, demographic models have increasingly shifted towards a finer-grained modeling of the population in the last decades. Understanding the aggregated dynamic is thus a major challenge brought by these non-linear “micro” models. One way of proceeding is to reduce the complexity of the aggregated population dynamic in order to obtain a tractable model.

With the rise of available data and computing power, so-called Microsimulation models have been developed in social sciences for the past decades (Li, J. and O ´Donoghue, C.(2013)). Used by government and institutional bodies11_{, they provide a simulation}

tool in order to address a broad variety of questions, ranging from evaluating the impact of policy changes and demographic shocks to the study of kinship structure. These models are mostly data-driven and their description often relies on a simulation algorithm. These features constitute an important limitation to their implementation, which needs a considerable amount of data (Silverman et al. (2011)). Furthermore, the complexity of microsimulation models can be significantly limited by computational costs. For instance, inter-individual interactions are often limited in microsimulation models, due to specification problems caused by data limitation or unobservable hidden processes (Zinn (2017)), as well as too high computational costs and time. Capturing the influence of “micro” behaviors at the aggregated level is also a difficult task, in a purely data-driven approach which does not allow for testing behavioral changes. A robust approach to the mathematical modeling of these complex dynamics could help us reach greater understanding on how heterogeneity and interactions operate at the macro-levels. In addition, this approach could also serve as a means to escape the tyranny of data described by Silverman et al. (2011), in order to alleviate “some of the burdens of the

time-consuming and combinatorially expensive data collection required to continue in the traditional fashion”.

11_{For instance, the MiCore tool have been developed as part of the European project Mic-Mac}

(2005-2009). Another widely used microsimulator is SOCSIM, which originates from a collaboration between Peter Laslett, Eugene Hammel and Kenneth Wachter.

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1.2 Motivations of the thesis

In very different fields, recent advances in probability, mathematical biology and ecology have contributed to the development of a new mathematical framework for individual-based stochastic population dynamics (see e.g. Champagnat et al. (2006);Fournier and

Méléard(2004); Méléard and Tran(2009)). These models have been applied for the study

of human population dynamics inBensusan(2010) andBoumezoued(2016). In particular, in order to model the dynamic evolution of heterogeneity inside a population, the latter included change of characteristics for individuals, such as changes of occupational class or marital status. However, these changes are described within a linear framework, while the behavior of individuals is often influenced by interactions with others. Thus, modeling the dynamic changes of the population composition within a non-linear framework and analyzing the aggregated dynamics produced by such models remains an important challenge.

Finally, the individual scale is not always the best chosen granularity for our purpose. Indeed, longitudinal data can be scarce and it can be sometimes more interesting to group individuals into larger risk classes. We will return to these questions in Chapter 2 and 3.

Guidelines of the thesis To sum up, let us state the main line of questioning that have served as general guidelines for this thesis:

• How can we define a general framework for the modeling of stochastic heterogeneous

populations? How does (socioeconomic) heterogeneity impact aggregated dynamics? What approximations can be made in order to reduce the complexity of the studied

evolution?

• What ingredients are needed for a more realistic theoretical modeling of human population dynamics, yielding populations with real-life characteristics?

• How can we provide an analytical and simulation framework serving as an experi-mental laboratory in order to support decision making? How can we test the validity or potential consequences of existing theories and common practices?

The complexity and the scope of these questions have led us to recognize the necessity of adopting an integrated approach, both theoretical and empirical, as neither theoretical modeling nor data seem to be sufficient to provide satisfactory explanations. On the one hand, theoretical modeling can be used to test the validity of some theories, when empirical studies can’t. Human social experiences are mostly non reproducible, and the

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challenging (National Research Council and Committee on Population(2011)). Moreover, theoretical modeling allows us to derive and/or justify approximations in order to simplify complex dynamics. On the other hand, the study of data throws light on important problems that need to be taken into account for a relevant modeling. The ability of data to point us in the right direction has been greatly increased in the current context of important data releases, which have in turn generated a need to provide a theoretical framework to study and deal with them. The fundamental interdisciplinary nature of the study of human populations prevents us from adopting too naive a modeling approach. Opportunities for mathematical developments arise in multiple directions when taking a cross-disciplinary approach, especially since theoretical models are able to incorporate qualitative data, for instance by defining specific interactions or behavior rules.

In this thesis, the modeling of heterogeneous population dynamics is addressed to from three different points of views: The first part (Chapter 2 and 3) is devoted to the modeling of non-linear changes of composition in a stochastic heterogeneous population model, and to the analysis of the aggregated dynamics produce by such models in presence of several timescale. The second part (Chapter 4) studies the impact over time of socioeconomic heterogeneity on aggregated longevity from an empirical point of view, based on recent English data by level of deprivation. The third and last part (Chapter 5) is a cross-disciplinary survey of selected topics on the evolution of human longevity and demographic models. The following sections give a overview of the specific motivations and result of each part of the thesis.

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1.3 Part I: Overview and summary of the results

1.3 Part I: Overview and summary of the results

The results presented in Chapter 2 and 3 have been obtained as part of an joint work with Nicole El Karoui.

1.3.1 Introduction

A number of empirical studies have highlighted diverging trends in lifespan since several decades, with growing differences within and among countries. If these inequalities are now well documented, the interpretation of empirical measurements made at different points in time is much more complex within this new heterogeneous framework. There is a real difficulty in comparing several subgroups at different points in time, due in part to time dependent phenomena changing dynamically the composition of the subgroups. For instance, while increases of geographical differences in health and mortality have been reported by several studies, it remains difficult to interpret mechanims underlying the observed inequalities: they could either be interpreted as true “trends” between different areas, or as the consequence of non observable changes of composition inside the areas, generated by changing patterns of internal migration. Dowd and Hamoudi(2014) take the example of migration between rural and urban areas in the US. Before the 70s, migrations from a rural to an urban county might have concerned more socioeconomically disadvantage individuals, while the reverse might have been true for individuals born during the post 1955 baby boom and moving during the late 70s/early 80s. Thus, these changing social patterning of migration could have generated an “artefactual trend” in mortality differentials between rural and urban areas.

Estimating flows of population between several areas or changes in the socioeconomic characteristics of a population is often a challenge. This may be due to a lack of information on incoming flows in cross-sectional data (we don’t know “where” people are coming from), or to sample size of longitudinal study that are to small to provide reliable estimates. Furthermore, individuals are not independent, and their behavior is influenced by interactions with others, resulting in the introduction of non-linearity or

density dependence in the population dynamics. For instance, the ability for individuals

to change of characteristics could be influenced by the number of individuals in the population. In that case, the composition of a particular area would be influenced not only by the social patterning of migration but also by the intensity of the incoming flow of individuals.

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understand how these changes affect the population and its longevity on a aggregated level. Indeed, the dynamic modeling of the population heterogeneity can produce unexpected or counter-intuitive effects at the aggregated level, which cannot be directly modeled by traditional “macro” demographic models.

The first part of this thesis focuses on the stochastic modeling of compositional changes in an heterogeneous population structured by discrete subgroups, and the study of the aggregated dynamic produced by such dynamics. Due to the complexity already introduced by changes of composition, the age-structure of the population is however not taken into account in the modeling. In this sens, the model does not focus particularly on human populations. For instance, similar models have been studied by Auger and coauthors in a deterministic framework, motivated by the study of spatial ecological systems (see e.g. Auger et al. (2000,2012);Marvá et al. (2013))12_.

In the classical framework of Markov multi-type Birth Death processes, heterogeneous populations are usually described by demographic events, that is by births (or entry) and deaths occurring in the subgroups. Here, changes in the composition of the population are also taken into account. They are described by so-calledswap events, corresponding to the move of an individual from one subgroup to another. Furthermore, the population evolution here is not assumed to be Markovian, in order to take into account additional randomness expressing the variability of the environment and time dependence (for instance the reduction of the mortality intensity over time). The heterogeneous population dynamics is thus called a Birth Death Swap (BDS) system.

These features lead us to adopt a point of view different than usual, by representing the population using multivariate counting processes on a state space different than the population state space. This point of view has proven to be very effective, as we detail in the outline below. In particular, in this representation, no specific assumptions are made to explain mechanisms generating swap or demographic events, so that the studied dynamics are very general.

The complexity generated by the presence of swap events makes it difficult to apprehend the population dynamic directly on an aggregated level. However, a separation of timescale can be observed between swap and demographic events in many cases. In particular, when changes in the composition of the population are fast in comparison with the demographic timescale, a simpler approximation of the aggregated population dynamics can be derived. The separation of timescale allows us to obtain an averaging result, reducing the Birth Death Swap system to an “autonomous” multi-type Birth Death process, with averaged intensities: due to swap events, non-linearities emerge at

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the aggregated level. Classical averaging results such as those obtained by Kurtz(1992)

or Yin and Zhang (2012) cannot be applied here. In order to overcome this difficulty, we

rely strongly on the pathwise representations of BDS system introduced in Chapter 2, as well as on the stable convergence of concerned processes.

Outline The general model is presented in the first section of Chapter 2. Our presen-tation gives an algebraic decomposition of the population, based on the multivariate process counting the number of events occurring in the population, called the jumps

counting system.

The second section is dedicated to the pathwise representation of the jumps counting system, as the solution of a multivariate Stochastic Differential System (SDE) driven by an extended Poisson measure. The existence of the jumps counting system is derived from a more general result obtained on the construction of multivariate counting systems by strong domination by a non-exploding process. The existence of BDS systems is then obtained as a simple corollary. The construction by strong domination offers several advantages. In particular, the existence of the jumps counting system is obtained under weaker assumptions than usual sub-linear growth or Lipschitz assumptions of the intensi-ties, and “free” tightness properties are derived from the construction in Chapter 3. In the last section of Chapter 2, an alternative construction of BDS systems is presented, called the Birth Death Swap decomposition algorithm. The decomposition algorithm allows us to disentangle swap events from demographic events, for instance when they are supposed to have their own timescale.

In Chapter 3, the BDS system is studied in the presence of two timescales, in which swap events are assumed to occur at a much faster timescale than demographic events. The model in presence of two timescales is described in the second section. In the third section of Chapter 3, a general identification result for the processes counting the number of demographic events is proven, in the limit when swap events become instantaneous with respect to demographic events. At the limit, the intensity of demographic events are averaged againststable limits of the population, when seen on a suitable space. This result is then applied in the last section, in which a convergence result is obtained in the particular case of deterministic swap intensities, but with general birth and death intensities. In particular, we show that the aggregated population converges to an au-tonomous (non Markov)non-linear Birth Death process, with birth and death intensities which have been averaged against stationary distribution of pure Swap processes. In order to prove the result, we heavily rely on the BDS decomposition algorithm of Chapter

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1.3.2 Pathwise constructions of Birth Death Swap systems

(Chapter 2)

Birth Death Swap systems

Let us introduce the stochastic model which serves as a basis for the results presented in Chapter 2 and 3. The population is structured in p discrete subgroups, and the state of the population is described at time t by the random vector Zt= (Zt1, .., Z

p

t) , where Zti

is the number of individuals in subgroup i. The temporal evolution of the population is thus described by the Np_{-valued càdlàg process Z = (Z}

t)t≥0, called the population

process.

As stated above, we consider two different types of events: demographic events - corre-sponding to a birth or death in a given subgroup - andswap events - corresponding to the move of an individual from one subgroup to another. To our knowledge, the terminology Birth Death Swap has been introduced in discrete time byHuber (2012), for the very different purpose of generating random variables as stationary distribution of pure jumps processes.

The evolution of the population is described by listing the events which change the composition of the population, rather than describing what happen to individuals. For instance, no information is given on the origin of a birth. A birth event can be either endogenous to the population - the individual has parents in the population whose characteristics determine his subgroup - or exogenous - the individual is an immigrant. This approach, quite different from the usual description of individual based models, has the advantage of providing important flexibility in the modeling. In particular, dynamics including strong non-linearity due to complicated interactions can be described very simply.

Jumps counting process In Section2.2, an algebraic representation of the population dynamic is given, based only on the study of the number of demographic and swap events. All processes are defined on a given probability space (Ω, G, P), equipped with equipped with a filtration (Gt) verifying the usual assumptions of right-continuity and

completeness.

We first define a unified description of the different types of events, particularly useful in the rest of the chapter. The set of all types of events is defined by a set J of cardinal

p(p + 1) (p birth events, p death events, and p(p − 1) swaps). With each type of event is

associated a corresponding jump ϕ(γ) of the population process:

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j and the corresponding jump is ϕ(b, j) = ej, where ej = (0, .., 1j, 0..).

- If γ = (d, i) is an event of type “death in subgroup i”, an individual is removed to subgroup i and the corresponding jump is ϕ(d, i) = −ei. Observe that this event cannot

occur if the subgroup is empty.

- Finally, if γ = (i, j), i ̸= j, is a swap event from i to j, an individual is simultaneously removed to subgroup i and added to subgroup j. This is also only possible if the subgroup

i is not empty, and the resulting change in the population is ϕ(i, j) = ej− ei.

As a càdlàg pure jump process, the population process Z can be written as the sum of its jumps and by distinguishing the different types of events we obtain that

Zt= Z0+ X γ∈J ϕ(γ)N_tγ, with N_tγ = X 0<s≤t 1{∆Zs=ϕ(γ)}. (1.3.1)

For each type of event γ ∈ J , the counting process Nγ _{is the process which counts the}

number of events of type γ which happened in the population. By assumptions, two events cannot occur at the same time and the processes Nγ _{have no common jumps.}

Thus, the process N = (Nγ₎

γ∈J indexed by J is a well defined multivariate counting

process, called the jumps counting process of Z. The previous affine relation can be rewritten as Zt= Z0 + ϕ ⊙ Nt. A matrix interpretation of ϕ is given in the beginning of

Chapter 2.

Actually, the jumps counting process is no other than a rewriting of the jump measure of

Z. This algebraic representation can thus be applied to any pure jump process generated

by a finite or countable type of jumps. For instance, a similar representation is used in

Anderson and Kurtz(2015) in the particular case of Continuous Time Markov Chains

for the modeling of chemical reaction network. However, this representation is to our knowledge less usual for population dynamics.

In many cases, the choice of time 0 for the initial condition Z0 is very arbitrary, and there

is a real interest in relaxing this condition. For instance, Massoulié (1998) defines the initial condition as the state at time 0 of some Np_{-valued process (ξ}

t)t≤0. In our setting,

we consider a generalized initial condition starting at a random date τ ≥ 0 from state

ζτ ∈ Np, defined by the entry process ξt= ζτ1{τ ≤t}. The population is then rewritten as, Zt= ξt+ ϕ ⊙ Nt, N

γ

t = P

τ <s≤t1{∆Zs=ϕ(γ)}

.

Population system Due to the many tools available for the study of counting processes, we are interested in the reverse approach, that is the construction of a population process from an entry process and a multivariate counting process. Given a couple (ξ, N), a

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not necessarily a well defined population process since its components can take negative values. A necessary and sufficient condition for the population process to be well-defined is actually the support condition (1.3.2), which ensures that no death or swap event can occur from an empty population (support condition), and that N does not increase on the set {t; ξt− = 0} (starting condition).

Definition 1.3.1 (Population system with random departure (Definition2.2.1)). a) Let (ξt= ζτ1{τ ≤t}) be an Np-valued entry process. A p(p + 1)-multivariate counting process N indexed by J is called a jumps counting process starting from ξ iff

   Starting condition 1{ξt−=0}dNt = 0 Support condition 1{ξi t−+(ϕ⊙N)it−=0}dN i,β t = 0 ∀i ∈ Ip, ∀β ∈ I(i) (1.3.2)

b) The companion population process of (ξ, N) is defined by Zt = ξt + ϕ ⊙ Nt. In

paticular, Z is a well-defined population process with the jumps counting process N and initial condition ξ. The triplet (ξ, N, Z) is called a population system.

The populations in which no demographic events occur are called swap processes and will play a very important role in the following. As for population systems, a swap system is defined by a triplet (ξ, Nsw, X), where the swap jumps counting process Nsw is now indexed by the set of swap events denoted by Jsw_{. The restriction of the function} ϕ to swap events is denoted by ϕs_{, and the companion swap process is defined by}

Xt = ξt+ ϕs⊙ Nswt .

Several useful transformations on population systems can be directly obtained from this algebraic representation of population systems. For instance a population system (Z0, N, Z) starting from 0 in state Z0 can be decomposed at a random time τ . The

population system stopped at time τ is (Z0, Nτt = Nt∧τ, Ztτ = Zt∧τ). We can also define

a population process starting from τ in state Zτ,

ξ_tτ(Z) = Zτ1{τ ≤t}, Nτt+ =

Rt

0 1]τ,∞)(s)dNs, Ztτ+ = ξtτ(Z) + ϕ ⊙ N τ+

t .

The population process can be decomposed in Z = Zτ _{+ ϕ ⊙ N}τ+ _{and N = N}τ_{+ N}τ+_.

Birth Death Swap intensity By describing the events changing the population composition rather than the behavior of individuals, a very flexible algebraic description of the population is obtained, based on the jumps counting system. Thanks to this general representation, Birth Death Swap systems are defined in the last part on the first section, by transferring the support conditions onto the intensity of jumps counting process. In order for the population system to become a Birth Death Swap system, additional assumptions are made on the multivariate intensity of the jump counting

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system.

Since only the companion population process Z is usually observed, the intensity process is assumed to be a functional of the population process rather than of the jump counting process itself. To go further and take into account some additional time-dependent randomness, the multivariate intensity process is also assumed to depend in a predictable way on additional randomness.

Definition 1.3.2 (BDS intensity functional and BDS system (Definition 2.2.2)). a) A BDS intensity functional µ(ω, t, z) = µ(t, z) = (µγ_{(t, z))}

γ∈J is a multivariate

G-predictable non-negative functional depending on z ∈ Np, satisfying

µ(t, 0) ≡ 0 and X

i∈Ip

X

β∈I(i)

µi,β(t, z)1{zi_=0}≡ 0, dt ⊗ dP a.s. (1.3.3)

b) A Birth Death Swap (BDS) system of intensity functional µ is a population system (ξ, N, Z)such that the jumps counting process N is a multivariate counting process of

Gt-intensity λt= µ(t, Zt−) = µ(t, ξt− + ϕ ⊙ N_t−).

Linear intensities: For each type of event γ, µγ_{(t, Z}

t−) is the intensity corresponding to the occurrence of the event of type γ inall the population, and not the rate at which the event of type γ can occur to one individual. A direct interpretation of intensities in term of individual rates can be derived in the case of linear intensities. For instance, if the death intensity in subgroup i is µ(d,i)_{(ω, t, z) = d}i_{(ω, t)z}i_{, the interpretation is}

that all individuals in subgroup i die independently at death rate di_{(ω, t). A similar}

interpretation can be given for linear swap or birth intensities.

Markov BDS: When the BDS intensity functional is an homogeneous deterministic

function µ(z), the BDS system is a Continuous Time Markov Chain (CTMC), and can be described using classical tools of CTMC.

Birth Death Swap Differential equation

In section 2.3 of Chapter 2, we tackle the issue of BDS systems and their pathwise realization. The two main results of this section are in the spirit of the recent renewed interest for pathwise representations of point processes, which has led to the development of a consequent body of literature in various domains. These representations are based on the pathwise realization of point processes as solutions of Stochastic Differential Equations (SDE) driven by Poisson measures. In particular, solutions are obtained from the thinning of an “augmented Poisson measure”.

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solution of multivariate SDEs driven by a Poisson measure. Focusing on the jumps counting process allows us to adopt the point of view of point processes, within a general framework similar to that ofMassoulié (1998).

Non-explosion is often central in the analysis of SDEs driven by Poisson measures, and in our setting, solutions are considered to be well-defined iff they stay finite in finite time with probability one.

The main point of the Section 2.3 is that the existence of BDS systems is derived from a more general result which is first obtained, on the construction of multivariate counting processes by strong domination by a non-exploding process. In particular, some of the usual assumptions on the intensity functional that are Lipschitz or sublinear growth conditions can be relaxed by using this result.

Construction of multivariate counting processes by strong domination

Theo-rem1.3.1 is the first main result of Section2.3, and concerns the construction of solutions

of SDEs driven by a Poisson measure by strong domination by a multivariate counting process driven by the same Poisson measure. Before stating the result, we give in this introduction a brief overview of the thinning of Poisson measure, which is described in more details in Subsection 2.3.1 of Chapter 2.

Given a multivariate Poisson measure ¯Q(dt, dθ) = (Qı_{(dt, dθ))}

ı=1...ρ with components of

intensity q(dt, dθ) = dt × dθ, and a ρ-multivariate predictable intensity process (¯λt), a

multivariate Cox process of Gt-intensity λt can be obtained bythinning and projection of

¯ Q, ¯Qλ t = Rt 0 R R+1{0<θ≤λs}Q¯(ds, dθ) = Rt

0Q¯(ds, ]0, λs]). Keeping the mark θ is sometimes

interesting, and a random measure ¯Q∆_{(dt, dθ) can be defined as the restriction of ¯}_Q

to the random set ∆ = {(t, θ); 0 < θ ≤ λt}: ¯Q∆(dt, dθ) = 1∆(t, θ)Q(dt, dθ). Q∆ is a

random measure of random intensity measure q∆_{(dt, dθ) = 1}

∆(t, θ) dt × dθ. When the

intensity ¯λ of the multivariate counting process ¯Qλ _{is a predictable functional of ¯}_Qλ _itself,

the thinning equation becomes a stochastic differential equation, driven by the Poisson measure ¯Q.

A ρ-multivariate counting process ¯Yα _{is said to}_{be strongly dominated by ¯Y}β_{, ¯}_Yα _{≺ ¯}_Yβ _iff

¯

Yβ_{− ¯}_Yα _{is a multivariate counting process, or equivalently iff all jumps of ¯}_Yα _{are jumps}

of ¯Yβ_{. In Theorem}_2.3.1_{, the solution ¯}_Yα _{(of intensity functional ¯}_{α(t, y)) of a multivariate}

SDE is built by strong comparison with a dominating process ¯Yβ _{(of intensity functional}

¯

β(t, y)), under the following assumption:

αı(t, ˜y) ≤ βı(t, ¯y), ∀1 ≤ ı ≤ ρ, ˜y ≤ ¯_{y ∈ N}ρ.

Theorem 1.3.1 ((Theorem 2.3.1)). Let ¯Q(dt, dθ) = (Qı_{(dt, dθ))}

ı∈E be a multivariate

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on Y = Nρ, where ¯α is assumed to be dominated by ¯β (¯α ≤ ¯β).

Assume the existence of a unique non-exploding solution ¯Yβ _{∈ N}ρ of the multivariate

SDE:

d ¯Y_tβ(¯y) = ¯Q(dt, ]0, ¯β(t, ¯y + ¯Y_t−β(¯y))]), (1.3.4)

Then, for all ˜y ≤ ¯y, there exists a unique (non-exploding) solution to the equation, d ¯Y_tα(˜y) = ¯Qdt, ]0, ¯α(t, ˜y + ¯Y_t−α(˜y))] (1.3.5)

Furthermore, ¯Yα_(˜_y) _{is strongly dominated by ¯Y}β_(¯_y)_{: ¯Y}α_(˜_{y) ≺ ¯}_Yβ_(¯_y)_.

Sketch of the proof: The thinning procedure is well-adapted to solve this problem, and

for Cox processes the answer is immediate. If two Cox processes Qλ_ti = Q(]0, t]×]0, λi_t]),

i = 1, 2 have ordered intensities λ1_t ≤ λ2

t, then the thinning construction using the same

Poisson measure for both processes directly yields that Qλ1 ≺ Qλ2

. The key to the proof is to rewrite Qλ1 as Qλ_t1 = Q(]0, t]×]0, λ1_t ∧ λ2

t]) = Q∆2(]0, t]×]0, λ1t]). This means that

Qλ1 can be obtained by thinning of Q∆2 _{instead of Q. In particular, all jump times of}

Qλ1 are jump times of Qλ2, and Qλ1 ≺ Qλ2 .

The direct application to general multivariate counting processes is not straightforward, since the order ¯α ≤ ¯β on the intensity functionals does not necessary imply a natural

order on the stochastic intensities α(t, ˜y + ¯Y_t−α) and β(t, ¯y + ¯Yt−β). The key idea of the

proof is however similar to the case of Cox processes. The idea is to study a slightly different version of (1.3.5), by replacing ¯Q with the measure ¯Q∆β_:

d ˜Y_tα= ¯Q∆β_{(dt, ]0, ¯}_{α(t, ˜}_{y + ˜}_Yα t−)]) = ¯Q(dt, ]0, ¯α(t, ˜y + ˜Y α t−) ∧ ¯β(t, ¯y + ¯Y β t−)]), (1.3.6) ˜

Yα is obtained by thinning of Q∆β_{, which guarantees the strong domination of ˜}_Yα _by

¯

Yβ. The existence and uniqueness of solutions of (1.3.6) is easier to prove, since the jump times of ¯Q∆β _{can be enumerated in increasing order, which is not the case for ¯}_Q_.

The proof is concluded by showing the equivalence of Equation (1.3.5) and (1.3.6). Theorem 1.3.1 generalizes the results of Rolski and Szekli (1991) for the comparison of Cox processes, and of Bhaskaran (1986) for the comparison of pure birth Markov processes. The proof of1.3.1 can actually be extended to the case of Point processes with continuous marks, and with intensity functional depending on the past of the process. This is the subject of a ongoing work with N. El Karoui.

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non-processes. The most simple example of such processes is the one dimensional Markov pure birth process, also called online Markov pure birth process, of intensity function here denoted by Kg(y). It is well-known that if the function g verifies the Feller criterion

∞

X

j=1

1

g(j) = ∞, (1.3.7)

the process does not explode in finite time. The multivariate case is an easy extension when the multivariate intensity function is a function of the size ¯y♮ =Pρ

1yı of the birth

process, K ¯g(¯y) = (Kgı(¯y♮)). Non-explosion is guaranteed if all functions gı satisfy the Feller criterion.

However, the domination by a multivariate Markov birth process is often not satisfactory. The assumption can be relaxed by using Cox Birth processes as dominating processes. Cox Birth processes are multivariate counting processes with product intensity ktg(¯¯ y♮).

They are obtained by replacing the constant K of Markov Births with a locally bounded predictable process (kt), i.e bounded by a sequence (Kp) along a nondecreasing sequence of

stopping times (Sp) going to ∞. The existence and non-explosion of Cox Birth processes

is a corollary of Theorem 1.3.1, since the solutions of (1.3.5) with intensity functional ¯

αp(t, ¯y) = (kt∧ Kp)¯g(¯y♮) are dominated by the Markov birth intensity ¯β(t, ¯y) = Kp¯g(¯y♮)

and do not depend on p on the interval [0, Sp].

Birth Death Swap multivariate SDE In the last part of the section, we come back to the study of BDS systems defined in1.3.2, and introduce the so-called Birth Death Swap multivariate SDE:

Definition 1.3.3 ( BDS multivariate SDE (Definition2.3.1)).

Let Q = (Qγ₎

γ∈J be a multivariate Poisson measure, µ(t, z) a BDS intensity functional

and (ξt) be an entry process. The Birth Death Swap multivariate SDE associated with

the entry process ξ and intensity functional µ is defined by

dNt= Q(dt, ]0, µ(t, ξt−+ ϕ ⊙ Nt−)]), with Zt= ξt+ ϕ ⊙ Nt. (1.3.8)

If N is a solution of (1.3.8), then (ξ, N, Z) is a BDS system of entry process ξ and BDS

intensity functional µ.

In order to obtain the existence and uniqueness of solutions (1.3.8), we give sufficient conditions on the birth intensities in order to obtain a solution of (1.3.8) by strong domination with a non-exploding process. In order to use the point process point of view of Theorem 1.3.1, the BDS intensity functional is expressed in terms of the jumps

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counting process, rather than in terms of the population process. Np_{-valued vectors are}

written as z = ξ + ϕ ⊙ ν, with ν = (νb_{, ν}d_{, ν}s_{) a vector indexed by J . The BDS intensity}

functional µ can be rewritten as functional of ν instead of z, λ(t, ν) = µ(t, ξt+ ϕ ⊙ ν).

The following assumption on the birth intensity is made:

Cox Birth Hyp ∀i ∈ Ip, λ(b,i)(t, ν) = µ(b,i)(t, ξt+ ϕ ⊙ ν) ≤ ktg(b,i)(ξt♮+ νb,♮) (1.3.9)

where ktgb = (ktg(b,i)) is a p-Cox birth intensity functional.

No further assumptions are made on the swap and death intensities, which are naturally dominated by an increasing function of the number of births:

Swap and death inequality λ(i,β)

(t, ν) = µ(i,β)(t, ξt♮+ ν) ≤ ˆµ(i,β)(t, ξ ♮

t + νb,♮) (1.3.10)

The dominating process is the p(p + 1) multivariate counting process G = (Gγ)γ∈J

defined in the following two steps:

The first step is to introduce the p-Cox Birth process Gt solution of the following

multivariate SDE:

dGb_t = Qb(dt, ]0, ktgb(ξ♮t+ G b,♮

t−)]). (1.3.11)

The second step consist in introducing “swap and death coordinates” to Gb, defined by the p and p(p − 1) multivariate Cox processes:

dGd_t = Qd(dt, ]0, ˆµ(t, ξ_t♮−+ G b,♮ t−)]), dGs_t = Qs(dt, ]0, ˆµ(t, ξ ♮ t−+ G b,♮ t−)]). (1.3.12) The(p(p + 1) multivariate counting process G = (Gb, Gd, Gs) is non-exploding and is called the dominating process.

Finally, the existence of the BDS multivariate SDE is obtained at the end of Section 2.3, under the Cox Birth domination assumption:

Theorem 1.3.2 (Theorem 2.3.2). Assume that the Cox Birth assumption (1.3.9) is

verified: µb_{(t, z) ≤ k}

tgb(z♮) where the components of gb are non-decreasing and satisfy

the Feller criterion (1.3.7). Moreover, assume that kt, µs(t, K) and µd(t, K) are locally

bounded in time for any K.

Then, there exists a unique solution to Equation (1.3.8),

dNt= Q(dt, ]0, µ(t, ξt−+ ϕ ⊙ Nt−)]), with Zt= ξt+ ϕ ⊙ Nt.

The triplet (ξ, N, Z) is a well-defined BDS system of BDS intensity functional µ and entry process ξ. Furthermore, N is strongly dominated by G, N ≺ G.

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BDS decomposition algorithm

There are many advantages in the construction by domination, in particular since the BDS jumps counting process N can be localized by a sequence of increasing stopping times which do not depend on the process itself. However, the same property can be a drawback when simulating the BDS system by strong domination. Indeed G can have much more jumps that N, making the simulation inefficient.

In Section 2.4 of Chapter 2, an alternative construction of the solution of (1.3.8) is presented, called the Birth Death Swap decomposition algorithm and based on the disentanglement of swap and demographic events. The decomposition algorithm is better suited to the simulation of BDS systems when swap and demographic intensities are of a very different nature, for instance when they are supposed to have their own timescale. The disentanglement of swap and demographic events will also be instrumental in the proof of Theorem 3.5.1 in Chapter 3.